As shown in the Appendix A, the most natural random field with logarithmic correlations corresponds to the so-called
Gaussian Free Field (GFF) in two spatial dimensions \(N=2\) , as well as its one-dimensional subsets.
It is one of the fundamental objects in physics and various issues of its statistics attracted
a lot of interest recently in conformal field theory, Schramm-Loewner evolution, and two-dimensional quantum gravity,
see e.g. some discussion in [1]}. Technically the problem of extracting the multifractality
exponents \(\tau _q\)  for the GFF amounts to ability to calculate efficiently
the disorder average of the free energy (REF ). Such task is in general considered to be one of the most difficult problems
in the statistical mechanics of systems with quenched disorder and we will not be able to perform such calculation explicitly
in \(N=2\)  GFF case Actually, in recent years some sophisticated probabilistic methods
were developed which allowed to address somewhat similar questions for GFF, see e.g. [2]} and the references therein. That development however
goes beyond the remit of the present lectures.. Instead, we are going
to outline such calculation for two particular choices of the models with logarithmically
correlated potentials where such calculation is indeed feasible. The
first model uses a special definition of distances between points in
space and is based on the idea of multiplicative
cascades which originated in the theory of turbulence, see e.g.
discussion and further references in [3]}. In fact, the
model is essentially equivalent to statistical mechanics of directed
polymers on disordered trees studied long ago in the seminal paper
by B. Derrida and H. Spohn [4]}.
Our second model will use standard Euclidean distances but exploits high
dimensionality of the embedding space: \(N\rightarrow \infty \) . Although the
details of the two models and the corresponding methods of solution
may look rather different, there is a general consensus that they address essentially the same physics: the so-called
freezing transition common to all disordered systems with logarithmic correlations. And indeed we shall see that the resulting
multifractality spectrum will be identical. In the final section we will give a short account of recent works on different
aspects of logarithmically correlated potentials.
